Instructional Week 6: February 8-12
Grade 8 ISTEP+
Focus Topic: Pythagorean Theorem
Paced Standards:
8.GM.8: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world
and other mathematical problems in two dimensions.+
8.GM.9: Apply the Pythagorean Theorem to find the distance between two points in a coordinate
plane.
PS: 1, 2, 3, 4, 5, 6, 7, and 8 +
Key Vocabulary
Pythagorean Theorem-The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the
squares of the other two sides.
Connection to other 8th Grade Standards:
8.GM.7: Use inductive reasoning to explain the Pythagorean relationship.
Prerequisite/Foundational Standards
7.NS.2: Understand the inverse relationship between squaring and finding the square root of a perfect
square integer. Find square roots of perfect square integers.
Teacher Background: Samples Problems for 8.GM.8
1. Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14
units. The figure below shows some of the dimensions but is not drawn to scale. (Illustrative Mathematics)
Is the shaded triangle a right triangle? Provide a proof for your answer.
Teacher Background Notes: Samples Problems for 8.GM.9
1. Using the Pythagorean Theorem, we can find the hypotenuse c of the smallest right triangle.
52+52=c2
Similarly, the hypotenuse of the right triangle with side lengths 7 and 14 is
If the shaded triangle is a right triangle, then the side-lengths must satisfy the Pythagorean Theorem. Since 75√ is the
longest of the three sides, it would be the hypotenuse, so if this is a right triangle, then the following equation must
be true:
However, looking at the left-hand side, we find that
and looking at the right hand side, we find that
and the equation in not true. So the shaded triangle is not a right triangle.
Resources for 8.GM.8

Scroll down this page and choose the activity for 8.G.7 called “Patterns in Prague.”
http://www.insidemathematics.org/index.php/8th-grade
 This tutorial helps explain how to use and apply the Pythagorean Theorem:
https://www.khanacademy.org/math/geometry/right_triangles_topic/pyth_theor/v/pythagorean-theorem-1
 This one minute video demonstrates an application of the Pythagorean Theorem:
http://www.teachertube.com/video/28238
Process Standards to Emphasize with Instruction
PS.1: Make sense of problems and persevere in solving them.
PS.2: Reason abstractly and quantitatively.
PS.3: Construct a viable argument and critique the reasoning of others.
PS.4: Model with mathematics.
PS.5: Use appropriate tools strategically.
PS.6: Attend to precision.
PS.7: Look for and make use of structure.
PS.8: Look for and express regularity in repeated reasoning.
Teacher Background: Samples Problems for 8.GM.9
1. This involves estimating the yardage off the diagram.
During the 2005 Divisional Playoff game between The Denver Broncos and The New England Patriots, Bronco
player Champ Bailey intercepted Tom Brady around the goal line (see the circled B). He ran the ball nearly all
the way to other goal line. Ben Watson of the New England Patriots (see the circled W) chased after Champ
and tracked him down just before the other goal line. (Illustrative Mathematics)
In the image below, each hash mark is equal to one yard: note also the field is 53 1/3 yards wide.
a.
How can you use the diagram and the Pythagorean Theorem to find approximately how many yards
Ben Watson ran to track down Champ Bailey?
b.
Use the Pythagorean Theorem to find approximately how many yards Watson ran in this play.
c.
Which player ran further during this play? By approximately how many more yards?
2. Doug is a dog, and his friend Bert is a bird. They live in Salt Lake City, where the streets are 1/16 miles apart
and arranged in a square grid. They are both standing at 6th and L. Doug can run at an average speed of 30
mi/hr through the streets of Salt Lake, and Bert can fly at an average speed of 20 mi/hr. They are about to
race to 10th and E.
a. Who do you predict will win, and why?
b. Draw the likely paths that Doug and Bert will travel.
c. What will you need to compare, in order to determine the winner?
d. Devise a plan to calculate these, without measuring anything.
e. Who will win the race?
Teacher Background Notes: Samples Problems for 8.GM.9
1. a. The triangle formed by Watson’s run, the sideline and the perpendicular line from Watson to the sideline,
make a right triangle. You could use the Pythagorean Theorem to find out approximately how far Watson
ran. You would have to estimate how much of the 160 feet (53 1/3 yard) width of the field represented his
vertical starting point. You can get a good estimate using the marks going along the end zone.
b. Looking at Ben Watson's position on the football field, it appears as if he has run between 90 and 92 yards
up the field in order to reach Champ Bailey at the one yard line. Counting the number of hashmarks at the
back of the end zone between Ben Watson's position and the top of the end zone, there appear to be
between 9 and 12. Disregarding the13 of a yard in the football field width (because Champ Bailey was within
the field when Watson tackled him), this means that Watson ran between 53-12=41 and 53-9= 44 yards
across the field. We can now apply the Pythagorean theorem to estimate how far Ben Watson has run: the
straight line path from Watson to the one yard line where he meets Bailey is the hypotenuse while the two
legs are a vertical line going from Watson's initial position down to the side line where he meets Bailey and a
horizontal line going all the way up the sideline where Bailey ran. For our lower end estimate for the
distance Watson has run, these legs are 41 yards and 90 yards respectively. Applying the Pythagorean
theorem gives
(90 yards)2+(41 yards)2=(run distance)2
8,100 yards2+1,681 yards2=(run distance)2
9781 yards2=(run distance)2
c. Using these lengths Watson’s run distance would be approximately 99 yards or about 297 feet.
Applying this same reasoning to obtain a high end reasonable estimate for the right triangle that models
Watson’s run, we get a right triangle with legs of 44 yards and 92 yards. This gives the following equation for
Watson's run:
(92 yards)2+(44 yards)2=(run distance)2
8,464 yards2+1,936 yards2=(run distance)2
10,400 yards2=(run distance)2
Using these lengths Watson’s run distance would be approximately 102 yards or about 306 feet. So a
reasonable estimate for the distance of Watson’s run would be between 99 and 102 yards or 297 and 306
feet.
The same process can be applied to find out how far Bailey has run though here the ambiguity in how far
Bailey has run across the field is far less because he starts off very close to the sideline. Champ Bailey started
from inside the Bronco’s end zone and ran to the Patriot’s 1-yard line. According to the picture, Bailey started
from roughly one to three yards behind the goal line in the end zone. He also started about three or four yards
from the sideline. Just as we did for Ben Watson, we can create a right triangle and use the hypotenuse to
model Bailey’s run: the legs for the run are the short line from Bailey's position in the end zone to the sideline
and the long line all the way up the field to the one yard line where he is tackled. For a low end estimate for
the length of this hypotenuse we use legs of lengths 3 yards and 100 yards respectively:
(100 yards)2+(3 yards)2=(interception run distance)2
10,000 yards2+9 yards2=(interception run distance)2
10,009 yards2=(interception run distance)2
Using these lengths Bailey’s run distance would be approximately 100 yards or 300 feet.
On the high end a good estimate for the length of the hypotenuse that models Bailey’s run would be:
(102 yards)2+(4 yards)2=(interception run distance)2
10,404 yards2+16 yards2=(interception run distance)2
10,420 yards2=(interception run distance)2
Using these lengths Bailey’s run distance would be approximately 102 yards or 306 feet. So our estimates
show that Bailey ran between 100 and 102 yards.
The range in possible run lengths for each player is similar. Both players ran roughly the same distance give
or take a yard. Depending on how each student measures or estimates the lengths of the triangle it is
possible that they might find that Watson ran a greater distance, that Bailey ran a greater distance or that
they ran the same distance. The focus here should be on the process of modeling the problem with right
triangles, how we came to the lengths of the right triangles, how we computed the distances as well as
communicating and critiquing the reasoning of others.
2. a. Either prediction is acceptable, as long as it’s justified. Bert, being able to fly directly, will travel the
shortest distance between the two points. However, his average speed is slower. Doug can travel faster, but
as he must stay on the ground, will have a farther distance to travel. (It is important for students to realize
that no matter the path Doug takes, as long as he only runs west and north, he will run 11 blocks: 7 blocks
west and 4 blocks north.)
b. I infer that Bert will take the shortest path possible, and draw a line segment between the start and finish to
represent his path. I can draw any combination of due-west and due-north 11-block paths for Doug.
(Drawing horizontally along 6th Ave and vertically along E St will be suggestive later for using the
Pythagorean Theorem. Select a student to share his/her drawing that has chosen this path.)
c. I’ll have to compare the time it takes Doug to run his path at his speed, and the time it takes Bert to fly his
path at his speed.
d. I need to know the distance that each travels. I already know Doug’s distance, and I can use the Pythagorean
theorem to calculate Bert’s distance. Once I know the distances, I can use d = rt to find their times. Solving d
= rt for time gives t = d/r. So I will divide each distance by the contestant’s rate to find the time it took him
to travel between the points. The contestant with the shorter time wins. Since the rates are given in miles
per hour, the result I calculate will be expressed in hours. It may be helpful to convert the unit of time to
minutes by multiplying the result by 60.
e. Doug’s distance = 11 blocks or 11/16 = 0.6875 miles. Bert’s distance (let x represent the length of Bert’s
direct path):
Bert’s distance is approximately 8.0623 blocks or 8.0623/16 = 0.5039 miles.
To find the time of travel, I’ll divide each distance by the corresponding rate.
Doug’s time = 0.6875 miles / 30 (mi/hr) = 0.023 hours or (0.023)(60) = 1.38 minutes
Bert’s time = 0.5039 miles / 20 (mi/hr) = 0.025 hours or (0.025)(60) = 1.51 minutes
Since Doug’s time is less than Bert’s time, Doug wins the race.
Resources for 8.GM.9

This tutorial from Khan Academy helps to explain concepts within this standard:
https://www.khanacademy.org/math/algebra/linear-equations- and-inequalitie/more-analyticgeometry/e/distance_formula

This interactive activity from Learn Zillion provides an opportunity for students to understand this standard:
https://learnzillion.com/lessons/1309-find-the-length-of-a-line-segment-on-the-coordinate-planeusing-the-pythagorean-theorem
Process Standards to Emphasize with Instruction
PS.1: Make sense of problems and persevere in solving them.
PS.2: Reason abstractly and quantitatively.
PS.3: Construct a viable argument and critique the reasoning of others.
PS.4: Model with mathematics.
PS.5: Use appropriate tools strategically.
PS.6: Attend to precision.
PS.7: Look for and make use of structure.
PS.8: Look for and express regularity in repeated reasoning.
Instructional Week 6 Assessment
ISTEP+ Grade 8
Name_____________________________
(8.GM.8)1. A kite with a 120-ft string attached to the ground is flying directly over the head is Seleana. Seleana is 24 ft
from where the kite is attached to the ground. If Seleana is 6 ft tall, how far above her head is the kite?
Round to the nearest foot.
a.
b.
c.
d.
112 feet
118 feet
122 feet
124 feet
(8.GM. 9)2. Jason has gone to the garden supply store to purchase a trellis to complete a garden he is landscaping. He
realizes that the blueprint for the job he is doing has torn and all he has left is the corner that shows the
location of the trellis. Fortunately a blueprint is a scale drawing so Jason can still figure out how much trellis
material he needs.
a. How can Jason use the blueprint to determine the length of the trellis?
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b. What is that length of the trellis? Show all work.
Instructional Week 6 Assessment
ISTEP+ Grade 8
(8.GM.8) 1. a
(8.GM.9) 2. 4 points total; 2 points each
a. Jason could draw a right triangle on his blueprint where the trellis is the hypotenuse. He can use the scale
and measure the length of each leg of the triangle. Then, solve the Pythagorean Theorem to find the
length of the trellis. Answers may vary.
b. 17 feet
All constructed response problems will be graded using the ISTEP+ Content Rubric.